Change in Momentum with Constant Acceleration: A Detailed Analysis

In the realm of physics, understanding the principles of motion and momentum is fundamental. This article delves into the calculation of the change in momentum of an object under constant acceleration, focusing on a specific example.

When a body is moving with a constant acceleration and starts from rest, the change in its momentum can be determined using simple yet effective equations of motion. Let's explore this scenario with the help of a practical example.

Understanding the Concept

The change in momentum of a body is given by the formula:

[text{Change in momentum} mDelta v]

where: m (mass) is measured in kilograms (kg). (Delta v) (change in velocity) is measured in meters per second (m/s).

Problem Statement and Known Quantities

A 5 kg body is moving with a constant acceleration of 5 m/s2 starting from rest. We need to find the change in momentum over a period of 4 seconds.

Calculation Steps

Step 1: Calculate the Final Velocity

We know the initial velocity u 0 m/s, the acceleration a 5 m/s2, and the time t 4 s. Using the equation of motion:

[v u at]

Substitute the values:

[v 0 5 times 4 20 , text{m/s}]

The final velocity of the body is 20 m/s.

Step 2: Calculate the Change in Velocity

The change in velocity is:

[Delta v v - u 20 - 0 20 , text{m/s}]

Step 3: Calculate the Change in Momentum

The change in momentum is given by:

[text{Change in momentum} m Delta v 5 times 20 100 , text{kg m/s}]

Therefore, the change in momentum in 4 seconds is 100 kg m/s.

Alternative Approach Using SUVAT Equations

Step 1: Identify the Relevant SUVAT Equation

For the equation of motion, we often use the SUVAT system. Here, we'll use the equation:

[v u at]

This is the most straightforward since it directly relates the final velocity with the initial velocity, acceleration, and time.

Step 2: Substitute the Given Values

Using the values u 0 m/s, a 5 m/s2, and t 4 s:

[v 0 5 times 4 20 , text{m/s}]

The final velocity is 20 m/s.

Step 3: Calculate the Change in Momentum

Using the change in momentum formula:

[Delta v 20 - 0 20 , text{m/s}]

Thus:

[text{Change in momentum} m Delta v 5 times 20 100 , text{kg m/s}]

Leveraging Other SUVAT Equations for Further Practice

Using the Equation (s ut frac{1}{2}at^2)

Let's consider a similar problem where we need to find the change in momentum using the distance covered. Assuming the final velocity is 20 m/s and the time is 4 s:

[s ut frac{1}{2}at^2 0 frac{1}{2} times 5 times (4^2) 40 , text{m}]

The distance covered is 40 meters.

Conclusion

Understanding and applying the principles of constant acceleration and the change in momentum is crucial for solving a variety of physics problems. By utilizing the equations of motion (SUVAT), we can efficiently determine the change in momentum for any given scenario.